Growth of a tessellation: Geometric rules drive development of stingray skeletal patterns

The skeletons of sharks and rays, fashioned from cartilage, and armored by a veneer of mineralized tiles (tesserae) present a mathematical challenge: How can the continuous covering be maintained as the skeleton expands? This study, using microCT and custom visual data analyses of growing stingray skeletons, systematically examines tessellation patterns and morphologies of the many thousand interacting tesserae covering the hyomandibula (a skeletal element critical to feeding), over a two-fold developmental change in hyomandibula length. The number of tesserae remains surprisingly constant, even as the hyomandibula expands isometrically, with all hyomandibulae displaying self-similar distributions of tesserae shapes/sizes. Although the distribution of tesserae geometries largely agrees with the rules for polyhedra tiling of complex surfaces—dominated by hexagons and a minor fraction of pentagons and heptagons, but very few other polygons—the agreement with Euler's classic mathematical laws is not perfect. Contrary to the assumed uniform growth rate (which is shown would create geometric incompatibilities), larger tesserae grow faster to accommodate skeletal expansion. It is hypothesized that this local regulation of global system complexity is driven by tension (from cartilaginous core expansion) in the fibers connecting tesserae, with strain-responsive cells orchestrating local mineral apposition.

鲨鱼和鳐鱼的骨骼，由软骨制成，并被矿物化瓷砖（马赛克）层覆盖，构成了一个数学难题：如何在骨骼扩张的过程中维持连续的覆盖？本研究利用微CT扫描和定制的视觉数据分析技术，对成长中的鳐鱼骨骼进行了系统研究，探讨了覆盖下颌骨（一种对摄食至关重要的骨骼结构）的数以千计的交互马赛克瓷砖的镶嵌模式和形态，并分析了下颌骨长度两倍发展变化中的这些模式。尽管下颌骨等比例扩张，马赛克瓷砖的数量却出人意料地保持恒定，所有下颌骨均显示出马赛克形状/大小的自相似分布。尽管马赛克几何形状的分布大致符合复杂表面多面体镶嵌的规则——以六边形为主，辅以少量五边形和七边形，但其他多边形却极为罕见——与欧拉经典数学定律的吻合并不完美。与假设的均匀生长速率（这会导致几何不兼容性）相反，较大的马赛克瓷砖生长速度更快，以适应骨骼的扩张。有假设认为，这种对全局系统复杂性的局部调节是由连接马赛克的纤维中的张力（来自软骨核心的扩张）驱动的，应变响应细胞在局部矿物沉积中起着协调作用。